Hasil nu penting jeung kawentar éta disebutna Teorema Central Limit Theorem; nu museurkeun kana variabel bébas nu pasipatan sebaranna bébas tur nilai eksptasi jeung varianna terhingga. Several generalizations exist which do not require identical distribution but incorporate some condition which guarantees that none of the variables exert a much larger influence than the others. Two such conditions are the Lindeberg condition and the Lyapunov condition. Other generalizations even allow some "weak" dependence of the random variables.

Consider the sum :Sn=X1+...+Xn. Then the expected value of Sn is nμ and its simpangan baku is σ n½. Furthermore, informally spéaking, the distribution of Sn approaches the normal distribution N(nμ,σ2n) as n approaches ∞.

In order to clarify the word "approaches" in the last sentence, we standardize Sn by setting

But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit théorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.

An equivalent formulation of this limit théorem starts with An = (X1 + ... + Xn) / n which can be interpreted as the méan of a random sample of size n. The expected value of An is μ and the standard deviation is σ / n½. If we normalize An by setting Zn = (An - μ) / (σ / n½), we obtain the same variable Zn as above, and it approaches a standard normal distribution.

Note the following apparent "paradox": by adding many independent identically distributed positive variables, one gets approximately a normal distribution. But for every normally distributed variable, the probability that it is negative is non-zero! How is it possible to get negative numbers from adding only positives? The réason is simple: the théorem applies to terms centered about the méan. Without that standardization, the distribution would, as intuition suggests, escape away to infinity.

The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit théorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions incréases without bound, under the conditions stated above.

Since the characteristic function of a convolution is the product of the characteristic functions of the densities involved, the central limit théorem has yet another restatement: the product of the characteristic functions of a number of density functions tends to the characteristic function of the normal density as the number of density functions incréases without bound, under the conditions stated above.

An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform.

There are some théorems which tréat the case of sums of non-independent variables, for instance the m-dependent central limit theorem, the martingale central limit theorem and the central limit theorem for mixing processes.